Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate

Abstract

Algebraic error-correcting codes that achieve the optimal trade-off between rate and fraction of errors corrected (in the model of list decoding) were recently constructed by a careful "folding" of the Reed-Solomon code. The "low-degree" nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes useful for list decoding arise out of the ArtinFrobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible M ∈ ‫ކ‬ q [T ]. The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case M = T ). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree 1 places that can be used for encoding.Our methods shed new light on algebraic codes and their list decoding, and lead to new codes with optimal trade-off between rate and error correction radius. Quantitatively, these codes provide list decoding (and list recovery/soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.

Bridging Shannon and Hamming: List Error-correction with Optimal Rate

“…Specifically, if a place P of a field K is inert in a finite Galois extension L/K with a place P in L above it, then the Frobenius automorphism Frob P satisfies Frob P (x) = x P (mod P ) for every x ∈ L that is regular at P , where P is the size of the residue field at P . Using this approach, we were able to extend the folded RS code construction to folded versions of certain algebraic-geometric codes based on cyclotomic function fields [20].…”

Division algebras and maximal orders for given invariants

“…Of course, there are other approaches to get a primitive element of L: in [1, Theorem 2.5], a normal integral basis of O L is given in terms of universal Gauss-Thakur sums. Alternatively the norm of a primitive h-torsion element can be taken [7,Theorem 4.3].…”

“…The approaches in [1,7] rely on computations in the field E, ours on computations in L /k F . In all three cases the coefficients of a minimal polynomial of a generator of L/F are computed.…”

Subspace designs based on algebraic function fields

“…This, however, runs into significant trouble, as the bound on number of solutions f to the functional equation analogous to (1), A 1 f + A 2 f σ + · · · + A s f σ s−1 = 0, is much higher. The list of solutions is either exponentially large and needs pruning via pre-coding the folded AG codes with subspace-evasive sets [GX12], or it is much bigger than q s−1 in the constructions based on cyclotomic function fields and narrow ray class fields where the folded AG codes work directly [Gur10,GX15].…”